code: def _rel_as_nonpos(constr, syms):
      """return `(np, d, aux)` where `np` is a list of nonpositive
      expressions that represent the given constraints (possibly
      rewritten in terms of auxilliary variables) expressible with
      nonnegative symbols, and `d` is a dictionary mapping a given
      symbols to an expression with an auxilliary variable. In some
      cases a symbol will be used as part of the change of variables,
      e.g. x: x - z1 instead of x: z1 - z2.
  
      If any constraint is False/empty, return None. All variables in
      ``constr`` are assumed to be unbounded unless explicitly indicated
      otherwise with a univariate constraint, e.g. ``x >= 0`` will
      restrict ``x`` to nonnegative values.
  
      The ``syms`` must be included so all symbols can be given an
      unbounded assumption if they are not otherwise bound with
      univariate conditions like ``x <= 3``.
  
      Examples
      ========
  
      >>> from sympy.solvers.simplex import _rel_as_nonpos
      >>> from sympy.abc import x, y
      >>> _rel_as_nonpos([x >= y, x >= 0, y >= 0], (x, y))
      ([-x + y], {}, [])
      >>> _rel_as_nonpos([x >= 3, x <= 5], [x])
      ([_z1 - 2], {x: _z1 + 3}, [_z1])
      >>> _rel_as_nonpos([x <= 5], [x])
      ([], {x: 5 - _z1}, [_z1])
      >>> _rel_as_nonpos([x >= 1], [x])
      ([], {x: _z1 + 1}, [_z1])
      """
      r = {}  # replacements to handle change of variables
      np = []  # nonpositive expressions
      aux = []  # auxilliary symbols added
      ui = numbered_symbols(
          "z", start=1, cls=Dummy
      )  # auxilliary symbols
      univariate = {}  # {x: interval} for univariate constraints
      unbound = []  # symbols designated as unbound
      syms = set(syms)  # the expected syms of the system
  
      # separate out univariates
      for i in constr:
          if i == True:
              continue  # ignore
          if i == False:
              return  # no solution
          if i.has(S.Infinity, S.NegativeInfinity):
              raise ValueError("only finite bounds are permitted")
          if isinstance(i, (Le, Ge)):
              i = i.lts - i.gts
              freei = i.free_symbols
              if freei - syms:
                  raise ValueError(
                      "unexpected symbol(s) in constraint: %s"
                      % (freei - syms)
                  )
              if len(freei) > 1:
                  np.append(i)
              elif freei:
                  x = freei.pop()
                  if x in unbound:
                      continue  # will handle later
                  ivl = Le(i, 0, evaluate=False).as_set()
                  if x not in univariate:
                      univariate[x] = ivl
                  else:
                      univariate[x] &= ivl
              elif i:
                  return False
          else:
              raise TypeError(
                  filldedent(
                      """
                  only equalities like Eq(x, y) or non-strict
                  inequalities like x >= y are allowed in lp, not %s"""
                      % i
                  )
              )
  
      # introduce auxilliary variables as needed for univariate
      # inequalities
      for x in syms:
          i = univariate.get(x, True)
          if not i:
              return None  # no solution possible
          if i == True:
              unbound.append(x)
              continue
          a, b = i.inf, i.sup
          if a.is_infinite:
              u = next(ui)
              r[x] = b - u
              aux.append(u)
          elif b.is_infinite:
              if a:
                  u = next(ui)
                  r[x] = a + u
                  aux.append(u)
              else:
                  # standard nonnegative relationship
                  pass
          else:
              u = next(ui)
              aux.append(u)
              # shift so u = x - a => x = u + a
              r[x] = u + a
              # add constraint for u <= b - a
              # since when u = b-a then x = u + a = b - a + a = b:
              # the upper limit for x
              np.append(u - (b - a))
  
      # make change of variables for unbound variables
      for x in unbound:
          u = next(ui)
          r[x] = u - x  # reusing x
          aux.append(u)
  
      return np, r, aux
  
imports: from sympy.core import sympify
  from sympy.core.exprtools import factor_terms
  from sympy.core.relational import Le, Ge, Eq
  from sympy.core.singleton import S
  from sympy.core.symbol import Dummy
  from sympy.core.sorting import ordered
  from sympy.functions.elementary.complexes import sign
  from sympy.matrices.dense import Matrix, zeros
  from sympy.solvers.solveset import linear_eq_to_matrix
  from sympy.utilities.iterables import numbered_symbols
  from sympy.utilities.misc import filldedent
label_desc: Use image node_id 7 for calling a global function with example usage: _rel_as_nonpos(constr, syms) and returns: np,  r,  aux, None, False